How is Pride and Prejudice a realist novel?
Pride and Prejudice can certainly be classified as a realistic novel.
We see the evidence of this in Austen's choice and treatment of topics, in the setting that she selects to develop her story, and in the type of characters that represent it.
Austen's topics are not far-fetched, nor fantastic: the novel is about the want for true love within a society in which marriage is a financially-bound institution.
This is far from a story about a damsel in distress; instead, it shows the reality of Austen's society within the circumstances of different women: the hysterically-concerned Mrs. Bennet, the realistic Elizabeth, the naive Jane, and the boy-crazy Lydia.
The settings of Pride and Prejudice are both simple and realistic: town and country in the form of London and the English countryside. Had Austen chosen a setting such as a castle, for example, the novel would have lost a lot of its realistic value.
Finally, the characters that Austen selects are quite realistic as well. We can see how the actions in which they engage affect each other. For instance, Elizabeth is quite aware that her family is embarrassing, and that her mother is close to intolerable. She even allows Darcy to say as much without trying to come in her defense.
Similarly, the way in which Austen describes Mr. Bennet's feelings for Mrs. Bennet before and after their marriage is far from romantic or fantastic:
[..] (Mr. Bennet) captivated by youth and beauty, and that appearance of good humour which youth and beauty generally give, had married a woman whose weak understanding and illiberal mind had, very early in their marriage, put an end to all real affection for her.
We see here a clear example of Austen's realistic treatment of how some marriages in her novel, especially the Bennet's marriage, can be construed as mistakes.
Respect, esteem, and confidence had vanished for ever; and all his views of domestic happiness were overthrown. But Mr. Bennet was not of a disposition to seek comfort, for the disappointment which his own imprudence had brought on, in any of those pleasures which too often console the unfortunate for their folly or their vice.
Therefore, Austen's selection of topics, settings, and characters are great evidence to demonstrate the realistic value of Pride and Prejudice.